"Be sure not to get lost" "Be sure not to trip going up the stairs." "Be sure not to stay too long." It is correct, just not used as often as the positive form.

Yes, it can be used that way. "Be careful while entering the cave. Just go and pick up the lamp. You will find treasures but be sure not to touch any of them. If you do, the cave will close and you will be stuck in for eternity" You can also say, "Be careful not to do sth" or "I want you not to do sth" etc. depending on context.